Finite Group Databasebeta

← C3×#83315 #11870699

Subgroups — C3×#83315 #11870699

Order 1152.

1011 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#532C211yesmaximal, characteristic
2C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
3C3×#5094C212yesmaximal, complemented
4C3×#5075C212yesmaximal, complemented
5C3×#5040C212yesmaximal, complemented
6C3×#5034C212yesmaximal, complemented
7C3×C4×#214C214yesmaximal, complemented
8C3×#4925C211yesmaximal, characteristic, complemented
9C3×#3103—31nomaximal
10#83315C311yesmaximal, characteristic, direct product factor
11C2×C32×#41—21no—
12C22×C3×Dic6C2211yescharacteristic
13C32×C4⋊Q8C2211yescharacteristic
14C32×#127C2211yescharacteristic
15C32×#124—21no—
16C32×#121—21no—
17C32×#121C2212yes—
18C32×C4×D4—21no—
19C32×C4×D4C2212yes—
20C2×C32×C42C2211yescharacteristic
21C2×C3×#214C2214yes—
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C2×C3×Dic3⋊C4—21no—
24C2×C3×C4×Dic3—21no—
25C3×C4×Dic6—21no—
26C3×C4×Dic6—21no—
27C3×C4×Dic6C2212yes—
28C3×C4×Dic6C2212yes—
29C3×#832C2214yes—
30C3×#830C2212yes—
31C3×#810C2214yes—
32C3×#808C2214yes—
33C3×C4⋊(C3⋊C8)C2212yes—
34C3×C4×C3⋊C8C2212yes—
35C3×#532—21no—
36C3×#532D311yescharacteristic
37C3×#532C2×C311yescharacteristic
38C2×C3×C4×Q8—31no—
39C2×C4×Dic6C2×C311yescharacteristic, complemented
40C3×#481—32no—
41C3×#475—32no—
42C3×#460—32no—
43C3×#448—32no—
44C3×C4×SD16—34no—
45C3×#431—31no—
46#5094C2×C312yescomplemented
47#5075C2×C312yescomplemented
48#5040C2×C312yescomplemented
49#5034C2×C312yescomplemented
50C4×#214C2×C314yescomplemented
51#4925C2×C311yescharacteristic, complemented
52C32×#41—44no—
53C2×C32×Q8—21no—
54C2×C32×D4—22no—
55C2×C32×D4—21no—
56C2×C32×D4—21no—
57C2×C32×D4C2312yes—
58C22×C32×C4—22no—
59C22×C32×C4D411yescharacteristic
60C22×C32×C4D411yescharacteristic
61C22×C32×C4C2311yescharacteristic
62C22×C3×Dic3—21no—
63C22×C3×Dic3—21no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—24no—
66C2×C3×Dic6C2312yes—
67C32×C4⋊C4—21no—
68C32×C4⋊C4C2312yes—
69C32×C22⋊C4—22no—
70C32×C22⋊C4—22no—
71C32×C42D411yescharacteristic
72C32×C42D411yescharacteristic
73C32×C42C2311yescharacteristic
74C32×C42C2311yescharacteristic
75C3×#214—28no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3C2312yes—
78C3×Dic3⋊C4—22no—
79C3×Dic3⋊C4—22no—
80C3×C4×Dic3—22no—
81C3×C4×Dic3—22no—
82C2×C3×C3⋊C8C2314yes—
83#3103—31noSylow
84C2×C3×#41—22no—
85C2×C3×#41—21no—
86C2×C3×#41—21no—
87C22×C3×Q8—31no—
88C22×Dic6C22×C311yescharacteristic
89C2×C3×SD16—34no—
90C3×C4⋊Q8—21no—
91C3×C4⋊Q8C2×D311yescharacteristic
92C3×C4⋊Q8C22×C311yescharacteristic
93C3×#127—21no—
94C3×#127C2×D311yescharacteristic
95C3×#127C22×C311yescharacteristic
96C3×#124—22no—
97C3×#124—21no—
98C3×#124—21no—
99C3×#121—22no—
100C3×#121—21no—
101C3×#121—21no—
102C3×#121—22no—
103C3×#121C2×D312yes—
104C3×#121C22×C312yes—
105C3×C4×Q8—32no—
106C3×C4×Q8—32no—
107C3×C4×Q8—61no—
108C3×C4×Q8—61no—
109C3×C4×D4—22no—
110C3×C4×D4—21no—
111C3×C4×D4—21no—
112C3×C4×D4—22no—
113C3×C4×D4C2×D312yes—
114C3×C4×D4C22×C312yes—
115C2×C3×C4⋊C4—31no—
116C2×C3×C4⋊C4—61no—
117C2×C3×C42—21no—
118C2×C3×C42—61no—
119C2×C3×C42C2×D311yescharacteristic
120C2×C3×C42C22×C311yescharacteristic
121C2×#214C22×C314yes—
122C2×C4⋊Dic3C22×C311yescharacteristic
123C2×Dic3⋊C4—21no—
124C2×C4×Dic3—21no—
125C4×Dic6—21no—
126C4×Dic6—21no—
127C4×Dic6C22×C312yes—
128C4×Dic6C22×C312yes—
129C3×#106—34no—
130C3×C4⋊C8—32no—
131C3×#103—34no—
132C3×#102—34no—
133C3×C22⋊C8—32no—
134C3×C4×C8—32no—
135#832C22×C314yes—
136#830C22×C312yes—
137#810C22×C314yes—
138#808C22×C314yes—
139C4⋊(C3⋊C8)C22×C312yes—
140C4×C3⋊C8C22×C312yes—
141C23×C32—22no—
142C23×C32C2×D411yescharacteristic
143C32×Q8—42no—
144C32×D4—24no—
145C32×D4—42no—
146C32×D4—44no—
147C32×D4—42no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—41no—
151C2×C32×C4—44no—
152C2×C32×C4SD1614yescomplemented
153C2×C32×C4C2×D411yescharacteristic
154C2×C32×C4C2×D411yescharacteristic
155C2×C32×C4C2×D411yescharacteristic
156C2×C32×C4C2×D411yescharacteristic
157C2×C32×C4C2×D411yescharacteristic
158C2×C32×C4C2411yescharacteristic
159C2×C3×Dic3—24no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—42no—
163C3×Dic6—24no—
164C3×Dic6—22no—
165C3×Dic6#4114yescomplemented
166C3×C3⋊C8—24no—
167#532C3×D311yescharacteristic
168C2×C4×Q8—31no—
169#481—32no—
170#475—32no—
171#460—32no—
172#448—32no—
173C4×SD16—34no—
174#431—31no—
175C3×#41—48no—
176C3×#41—44no—
177C3×#41—44no—
178C2×C3×Q8—22no—
179C2×C3×Q8—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—32no—
182C2×C3×Q8—64no—
183C2×C3×Q8—61no—
184C2×C3×D4—22no—
185C2×C3×D4—24no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—22no—
194C2×C3×D4C22×D312yes—
195C2×C3×D4C23×C312yes—
196C22×C3×C4—24no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—61no—
203C22×C3×C4—61no—
204C22×C3×C4#6711yescharacteristic
205C22×C3×C4#6711yescharacteristic
206C22×C3×C4C3×D411yescharacteristic
207C22×C3×C4C3×D411yescharacteristic
208C22×C3×C4C22×D311yescharacteristic
209C22×C3×C4C23×C311yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—21no—
212C2×Dic6—21no—
213C2×Dic6—24no—
214C2×Dic6C23×C312yes—
215C3×SD16—68no—
216C2×C3×C8—34no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—32no—
222C3×C4⋊C4—62no—
223C3×C4⋊C4—62no—
224C3×C4⋊C4—61no—
225C3×C4⋊C4C22×D312yes—
226C3×C4⋊C4C23×C312yes—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—62no—
238C3×C42—62no—
239C3×C42#6711yescharacteristic
240C3×C42#6711yescharacteristic
241C3×C42C3×D411yescharacteristic
242C3×C42C3×D411yescharacteristic
243C3×C42C22×D311yescharacteristic
244C3×C42C22×D311yescharacteristic
245C3×C42C23×C311yescharacteristic
246C3×C42C23×C311yescharacteristic
247#214—28no—
248C4⋊Dic3—21no—
249C4⋊Dic3C23×C312yes—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252C4×Dic3—22no—
253C4×Dic3—22no—
254C2×C3⋊C8C23×C314yes—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—42no—
258C22×C32—42no—
259C22×C32—42no—
260C22×C32C2×SD1612yes—
261C22×C32C22×D411yescharacteristic
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—42no—
266C32×C4C2×SD1612yes—
267C32×C4C2×SD1612yes—
268C32×C4C2×#4112yes—
269C3×Dic3—24no—
270C3×Dic3—42no—
271C2×#41—21no—
272C22×Q8—31no—
273C2×SD16—34no—
274C4⋊Q8C2×C3×D311yescharacteristic
275#127C2×C3×D311yescharacteristic
276#124—21no—
277#121—21no—
278#121C2×C3×D312yes—
279C4×Q8—32no—
280C4×Q8—32no—
281C4×Q8—61no—
282C4×Q8—61no—
283C4×D4—21no—
284C4×D4C2×C3×D312yes—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—61no—
287C2×C42—61no—
288C2×C42C2×C3×D311yescharacteristic
289#106—34no—
290C4⋊C8—32no—
291#103—34no—
292#102—34no—
293C22⋊C8—32no—
294C4×C8—32no—
295C23×C3—24no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3C2×#6711yescharacteristic
300C23×C3C2×C3×D411yescharacteristic
301C3×Q8—34no—
302C3×Q8—44no—
303C3×Q8—42no—
304C3×Q8—42no—
305C3×Q8—64no—
306C3×Q8—62no—
307C3×D4—24no—
308C3×D4—24no—
309C3×D4—44no—
310C3×D4—44no—
311C3×D4—42no—
312C3×D4—42no—
313C3×D4—44no—
314C3×D4—48no—
315C3×D4—44no—
316C3×D4—44no—
317C3×D4—42no—
318C3×D4—42no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—24no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—48no—
336C2×C3×C4—44no—
337C2×C3×C4—44no—
338C2×C3×C4—64no—
339C2×C3×C4—62no—
340C2×C3×C4—62no—
341C2×C3×C4—122no—
342C2×C3×C4#21414yescomplemented
343C2×C3×C4C3×SD1614yescomplemented
344C2×C3×C4C2×#6711yescharacteristic
345C2×C3×C4C2×#6711yescharacteristic
346C2×C3×C4C2×#6711yescharacteristic
347C2×C3×C4C2×#6711yescharacteristic
348C2×C3×C4C2×#6711yescharacteristic
349C2×C3×C4C2×C3×D411yescharacteristic
350C2×C3×C4C2×C3×D411yescharacteristic
351C2×C3×C4C2×C3×D411yescharacteristic
352C2×C3×C4C2×C3×D411yescharacteristic
353C2×C3×C4C2×C3×D411yescharacteristic
354C2×C3×C4C23×D311yescharacteristic
355C2×C3×C4C24×C311yescharacteristic
356C2×Dic3—24no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—42no—
360Dic6—24no—
361Dic6—22no—
362Dic6C3×#4114yescomplemented
363C3×C8—64no—
364C3⋊C8—24no—
365C2×C32—22no—
366C2×C32—82no—
367C2×C32#52511yescharacteristic
368C2×C32C22×SD1611yescharacteristic
369C2×C32#57511yescharacteristic
370#41—44no—
371C2×Q8—21no—
372C2×Q8—32no—
373C2×Q8—64no—
374C2×Q8—61no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4C22×C3×D312yes—
379C22×C4—22no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4C3×#6711yescharacteristic
383C22×C4C3×#6711yescharacteristic
384C22×C4C22×C3×D311yescharacteristic
385SD16—68no—
386C2×C8—34no—
387C4⋊C4—21no—
388C4⋊C4—32no—
389C4⋊C4—61no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4C22×C3×D312yes—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C42—62no—
396C42—62no—
397C42C3×#6711yescharacteristic
398C42C3×#6711yescharacteristic
399C42C22×C3×D311yescharacteristic
400C42C22×C3×D311yescharacteristic
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—22no—
406C22×C3—21no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3—44no—
410C22×C3—44no—
411C22×C3—44no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3C2×#21412yes—
419C22×C3C2×C3×SD1612yes—
420C22×C3C22×#6711yescharacteristic
421C22×C3C22×C3×D411yescharacteristic
422C3×C4—22no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4—44no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—64no—
438C3×C4—122no—
439C3×C4C2×#21412yes—
440C3×C4C2×#21412yes—
441C3×C4C2×C3×SD1612yes—
442C3×C4C2×C3×SD1612yes—
443C3×C4D3×#4112yes—
444C3×C4C2×C3×#4112yes—
445Dic3—24no—
446Dic3—42no—
447C32#310311yescharacteristic, complemented, Sylow
448C23—22no—
449C23C2×C3×#6711yescharacteristic
450Q8—34no—
451Q8—42no—
452Q8—64no—
453Q8—62no—
454D4—24no—
455D4—42no—
456D4—44no—
457D4—42no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—64no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—122no—
466C2×C4C3×#21414yescomplemented
467C2×C4C2×C3×#6711yescharacteristic
468C2×C4C2×C3×#6711yescharacteristic
469C2×C4C2×C3×#6711yescharacteristic
470C2×C4C2×C3×#6711yescharacteristic
471C2×C4C2×C3×#6711yescharacteristic
472C2×C4C23×C3×D311yescharacteristic
473C8—64no—
474C2×C3—22no—
475C2×C3—22no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—42no—
480C2×C3—84no—
481C2×C3—82no—
482C2×C3—82no—
483C2×C3C22×#21411yescharacteristic
484C2×C3#582211yescharacteristic
485C2×C3#582811yescharacteristic
486C2×C3C3×#52511yescharacteristic
487C2×C3C22×C3×SD1611yescharacteristic
488C2×C3C3×#57511yescharacteristic
489C22—21no—
490C22—21no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22C2×C3×#21412yes—
495C22C22×C3×#6711yescharacteristic
496C4—21no—
497C4—21no—
498C4—21no—
499C4—42no—
500C4—64no—
501C4—122no—
502C4C2×C3×#21412yes—
503C4C2×C3×#21412yes—
504C4C3×D3×#4112yes—
505C3—21no—
506C3#8331511yescharacteristic, direct product factor
507C3C3×#310311yescharacteristic, complemented
508C2—22no—
509C2—82no—
510C2C22×C3×#21411yescharacteristic
511C2C3×#582211yescharacteristic
512C2C3×#582811yescharacteristic
subgroupG/H
centerC2×C3×C4C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC32×#532C2
socleC22×C32C22×D4
hypercenterC3×#532D3
hypocenterC3C3×#3103
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ C3×#532

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×C4×Dic6C2semidirect
3. C3×#5094C2semidirect
4. C3×#5075C2semidirect
5. C3×#5040C2semidirect
6. C3×#5034C2semidirect
7. C3×C4×#214C2semidirect
8. C3×#4925C2semidirect
10. #83315C3direct (N × H)
39. C2×C4×Dic6C2×C3semidirect
46. #5094C2×C3semidirect
47. #5075C2×C3semidirect
48. #5040C2×C3semidirect
49. #5034C2×C3semidirect
50. C4×#214C2×C3semidirect
51. #4925C2×C3semidirect
152. C2×C32×C4SD16semidirect
165. C3×Dic6#41semidirect
342. C2×C3×C4#214semidirect
343. C2×C3×C4C3×SD16semidirect
362. Dic6C3×#41semidirect
447. C32#3103semidirect
466. C2×C4C3×#214semidirect
506. C3#83315direct (N × H)
507. C3C3×#3103semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C32×#5322
C2×C3×C4×Dic62
C3×#50942
C3×#50752
C3×#50402
C3×#50342
C3×C4×#2142
C3×#49252
C3×#31033
#833153